VARIATIONAL METHOD IN RELATIVISTIC QUANTUM FIELD THEORY WITHOUT CUTOFF - MPG.PURE
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Variational method in relativistic quantum field theory without cutoff
Antoine Tilloy∗
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany and
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München
The variational method is a powerful approach to solve many-body quantum problems non per-
turbatively. However, in the context of relativistic quantum field theory (QFT), it needs to meet
3 seemingly incompatible requirements outlined by Feynman: extensivity, computability, and lack
of UV sensitivity. In practice, variational methods break one of the 3, which translates into the
need to have an IR or UV cutoff. In this letter, I introduce a relativistic modification of continuous
matrix product states that satisfies the 3 requirements jointly in 1 + 1 dimensions. I apply it to the
self-interacting scalar field, without UV cutoff and directly in the thermodynamic limit. Numerical
evidence suggests the error decreases faster than any power law in the number of parameters, while
arXiv:2102.07733v1 [quant-ph] 15 Feb 2021
the cost remains only polynomial.
Introduction – Quantum field theory (QFT) lies at adjust the state parameters towards fitting shorter and
the root of fundamental physics, and is the most fun- shorter distances, paradoxically degrading the accuracy
damental approach we so far have to understand mi- at physically relevant length-scales. According to Feyn-
croscopic phenomena. A vexing problem of theoretical man, only Gaussian states could fit these 3 requirements,
physics is that QFTs are rarely ever solvable. We seem which excluded the variational method for interacting
to know the rules of particle physics, at least to a good theories.
precision, but hardly know what they give in general. Modern variational approaches swallow at least one of
Until recently, there were essentially two approaches to Feynman’s bullets. Hamiltonian truncation (HT) and its
deal with QFT approximately: perturbation theory [1] renormalized refinements [5] use a vector space, the free
and lattice Monte-Carlo [2, 3]. The first provides results Fock space, as state manifold. With an IR cutoff, en-
without cutoff in momenta, valid “all the way down” for ergy levels get discretized, and there is only a finite num-
a true QFT, but accurate only for small coupling. The ber of basis states under a truncation energy ET . With
second works at strong coupling, but introduces a short these two cutoffs, the energy minimization is a simple fi-
(UV) and long (IR) distance cutoff. nite dimensional linear problem. On the other hand, HT
Variational methods are a seducing third way. The clearly breaks extensivity, as the number of basis states
idea is to put forward a manifold M of quantum states and thus parameters grows exponentially as the system
|ψw i, specified by a small number of parameters w, and size (IR cutoff) is increased. Because the prefactors are
to minimize the expectation value of the Hamiltonian H favorable, and extrapolations reliable [6], HT can still be
over these parameters. a precise method in practice [7–10].
hψw |H|ψw i Tensor network states [11, 12], defined on the lattice,
|groundi ' |ψw i for w = argmin (1) are a naturally extensive class of states. In their 1 + 1 di-
M hψw |ψw i
mensional incarnation, the matrix product states (MPS)
If the manifold is guessed right, this can provide a good [13], local observables are also efficiently computable. In
non-perturbative approximation to the ground state, 2010, Cirac and Verstraete took the continuum limit of
from which one may then compute observables. MPS, to get the continuous MPS (CMPS) [14]. While it
In the context of relativistic QFT, the variational provides an efficient ansatz for non-relativistic QFT, it
method was submitted to a devastating criticism by still suffers from Feynman’s third objection in the rela-
Feynman in 1987, who listed 3 crippling objections, or tivistic context.
rather requirements on the state manifold that could not In practice CMPS can still be used for relativistic QFT,
possibly be met jointly [4]. The first is extensivity: the but one needs to add a UV cutoff in the Hamiltonian, that
states should be extensive, in the sense that increasing acts as Lagrange multiplier to prevent the state from even
the system size (IR cutoff) should increase the dimension fitting the UV [15, 16]. This limits the range of validity
of the manifold at most linearly (and not exponentially). of the results, breaks the strict variational nature of the
The second is computability: the states should be such approach, and partially defeats the purpose of going to
that expectation values of local observables can be com- the continuum in the first place. This UV difficulty is
puted reasonably efficiently. This is needed to minimize understandable: at short distances, true 1+1d relativistic
the energy, but also to extract physical predictions once QFT are conformal field theories, valid all the way down,
the state is known. The last requirement is specific to rel- without cutoff scale. CMPS cannot capture this short
ativistic QFT, and is a lack of sensitivity to UV features. distance behavior by construction, as they are regular at
Since the energy density of relativistic QFT is dominated short distances. The necessity of a UV cutoff thus seems
by arbitrarily large momenta, minimizing the energy will inevitable.2
This situation is frustrating because, at least for super- 22], matrix product states [23, 24], resummed perturba-
renormalizable and even asymptotically free theories, the tion theory [25]. All these methods, apart from pertur-
UV behavior CMPS fail to capture is otherwise trivial. bation theory, require at least one cutoff, UV or IR, and
Can we not include this “free” behavior exactly? My thus extrapolations.
objective in this letter is to show that in d = 1 + 1 di- The state manifold – A RCMPS is a quantum state
mensions, this is possible. The requirements of Feynman belonging to the free Fock space, parameterized by 2 D ×
can be jointly satisfied: one can put forward an exten- D complex matrices Q, R and defined as
sive and efficiently computable class of states, that comes
without cutoff (UV or IR), and that gives the energy den- Z
sity and all local observables to arbitrary precision upon |Q, Ri = tr P exp dx Q ⊗ 1 + R ⊗ a† (x) |0ia .
optimization. (5)
This new class of states, the relativistic CMPS In this formula, the trace is taken over the finite D di-
(RCMPS), borrow most of their definition from CMPS. mensional auxiliary space of matrices, P exp is the path-
The new ingredient is a change of operator basis and ordered exponential, and a† (x) is a creation operator such
Fock space (Bogoliubov transform), that provides the that [a(x), a† (y)] = δ(x − y)1. The state |0ia is the Fock
right large momentum behavior. This new Fock space vacuum annhilitated by all the a(x). The bounds in the
is the Fock space adapted to the free part of the the- integral can be an interval [−L, L] or R, that is directly
ory, which is precisely the one used in the Hamiltonian the thermodynamic limit, which makes extensivity mani-
truncation approach (up to a removal of the IR cutoff). fest. The bond dimension D is a proxy the expressiveness
RCMPS can thus be seen as a hybridization of HT and of the state manifold: the larger it is, the more knobs one
tensor network methods. can tune to lower the energy and fit the true ground state.
Defining RCMPS and obtaining their basic properties The definition (5) would be that of a standard CMPS
is rather straightforward and done in the present let- if a† (x) were chosen to be a local creation operator asso-
ter. However, evaluating the energy expectation value ciated to the conjugated pair φ(x), π(x), i.e.
for a given theory and then minimizing it by varying p canonically √
ψ † (x) = ν/2φ(x) − i( 2ν)−1 π(x) for some ν. This
the state parameters, requires slightly lengthier compu- would be a natural non-relativistic choice, preserving lo-
tations. They are presented in full glory in a companion cality, but creating the UV difficulties we discussed pre-
paper [17], that also provides more in depth discussions. viously.
The model – The prototypical theory we will apply
Instead, as the notation suggests, I take a(x) to be the
RCMPS to is the self-interacting scalar, a.k.a. φ42 theory.
Fourier transform of ak
The model is specified by its Hamiltonian
Z
Z 2
π (∂x φ)2 m2 2
1
H =: + + φ + g φ4 :a . (2) a(x) = dk eikx ak . (6)
2π
R 2 2 2
The normal-ordering is done with respect to the opera- This operator rarely ever appears in relativistic QFT. It
tors ak , a†k that diagonalize the free part of the Hamilto- is not local in the fields φ, π because of the (lack of) factor
√
nian obtained for g = 0. More precisely, the field opera- ωk . As discussed in the companion paper, it is actually
tors admit the mode expansion not the only choice, but it is arguably the simplest to
r make the ansatz match the UV behavior of the QFT.
Z
1 1 ikx
Indeed, if R and Q are zero, the RCMPS is just the
φ(x) = dk e ak + e−ikx a†k (3)
2π 2 ωk Fock vacuum |0ia , which is the ground state at g = 0.
It thus has the the right short distance behavior for free,
Z r
1 ωk ikx
π(x) = dk e ak − e−ikx a†k , (4) without the need for any parameter tuning. When R, Q
2π 2
are non-zero, the state lies in the same Fock space and the
√ UV behavior remains unchanged. One can further prove
where ωk = m2 + k 2 and [ak , a†k0 ] = 2πδ(k − k 0 ). Cru-
cially, in 1 + 1 dimensions, the normal-ordering : :a that local expectations values, like the energy density,
is sufficient to renormalize all the UV divergences (cor- are finite and well behaved [17].
responding to perturbation theory tadpoles), and H is Computations, in a nutshell – Since a(x) verifies the
a legitimate self-adjoint operator [18, 19]. While easy to same commutation relations as ψ(x), the standard CMPS
define, φ42 theory is surprisingly difficult to solve. It is not formulas [26], which depend only on this algebra, can
integrable, and its behavior at strong coupling g & m2 is be reused in the RCMPS context. In particular, all lo-
challenging to probe numerically. cal normal-ordered correlations functions of a(x) have a
This model has been studied with a wide variety of compact algebraic expression as a trace over finite dimen-
methods: renormalized Hamiltonian truncation [5, 8, 9], sional matrices.
Monte-Carlo [20], tensor network renormalization [21, This is seen by introducing the generating functional3
Zj 0 ,j : energy density hhiQ,R relative error
hQ, R| exp j 0 a† exp j a |Q, Ri
R R 0.0
Zj 0 ,j = , (7)
hQ, R|Q, Ri
which can be used to compute all normal-ordered correla- −0.5
g=2 10−1
tion N -point functions of a(x), a† (x) by taking functional RHT
derivatives. One can show [26] that this generating func- D=4
tional has an exact expression: −1.0 D=6
g=1
Z D=9
0
Zj ,j = tr P exp
0 dx T + j(x)R ⊗ 1 + j (x)1 ⊗ R̄ 10−2
0 2 4 3 5 7 9
(8)
g D
where T = Q ⊗ 1 + 1 ⊗ Q̄ + R ⊗ R̄ is the transfer op-
erator and the trace is taken over the tensor product of
two copies of the original D dimensional auxiliary Hilbert FIG. 1. Left: Approximate ground state energy density as a
space. For example, on the interval [−L, L], and for function of the coupling g for m = 1, compared with the RHT
L ≥ x ≥ y ≥ −L this gives: results of [5]. Right: relative error in the energy density as a
function of the bond dimension D, taking the RHT results in
h i [9] for g = 1 and g = 2 as close to exact comparisons.
ha† (x)a(y)i = tr e(L−x)T (1 ⊗ R̄)e(x−y)T (R ⊗ 1)e(y+L)T ,
(9)
and other correlation functions take a similar form. This could use whatever numerical solver, e.g. based on quasi-
formula can be further simplified in the thermodynamic Newton methods. It turns out that for the minimization
limit by making a proper choice of gauge [17]. to be efficient, it is better to use a more elaborate tan-
We are not yet done if we want to evaluate the energy gent space approach [28] and implement fast approximate
density. The latter is local in the fields φ, π which are imaginary time evolution [17]. Again, all that matters is
not local in a, a† , e.g. that it can be done efficiently.
Z r Results – The results for the ground state energy den-
1 1 ikx
φ(x) = dk e ak + e−ikx a†k sity of φ42 are shown in Fig. 1 and compared with the
2π 2 ωk
Z renormalized Hamiltonian truncation (RHT) computa-
1 dk dy ik(x−y)
= √ e a(y) + e−ik(x−y) a† (y) tions of [5]. Even a very moderate D = 4, corresponding
2π 2 ωk to 32 independent real parameters, provides qualitatively
Z
accurate results. For g = 1 and g = 2, the RHT extrapo-
= dy G(x − y)a(y) + Ḡ(x − y)a† (y) (10)
lations in [9] can be trusted, and are sufficiently close to
the exact values to evaluate the RCMPS error. Computa-
where G(x) is a smooth kernel away from x = 0, which, tions up to D = 9 suggest that it decreases approximately
crucially, decays exponentially as |x| → +∞, with a exponentially as a function of D.
rate proportional to m. Rewriting the Hamiltonian den-
For larger values of the coupling g ≥ 3, the RCMPS
sity in terms of a(x), a† (x) thus yields as many integrals
give substantially lower values for the energy density than
as the degree in the fields. This lack of strict locality
RHT results from [5]. Since the RCMPS results are rig-
is a technical inconvenience but it is not expected to
orous upper bounds, it means its energies are closer to
make the RCMPS less good at approximating the ground
the true one.
state [27].
Local observables can be straightforwardly computed
Ultimately, expressing the Hamiltonian density h as an
once the ground state is known, just like Feynman or-
integral of a’s and then using the exact expressions for
dered, since correlation functions have an explicit form.
expectation values of a’s, one can write
As an illustration, the normal-ordered momentum space
hhiQ,R = f (Q, R) (11) two-point function h: φp φq :iQ,R = δ(p + q)D(p) is shown
in Fig. 2. Results converge as a function of the bond
where f involves nested integrals of traces of matrices. dimension, at least deep in the symmetric and symmetry
This function f is finite, but takes a complicated form, broken phase. Getting near the critical point gc ' 2.77
which is derived in the companion paper [17]. What mat- [22] from the symmetric side, results are less accurate as
ters for us here is that f can be efficiently evaluated expected: RCMPS with low values of D (here 4, 6) fall
numerically with a polynomial cost in D (in practice, in the symmetry broken phase and the RCMPS is in the
∝ D4 ). right phase only for larger values of D (here D = 9).
All that remains is to minimize over R and Q to find Note that since the QFT is relativistic, correlation func-
an approximation of the ground state. In principle, one tions at equal time already provide a partial access to4
g=1 g = 2.7 g=4 relativistic QFT and require a UV cutoff to deal with
relativistic ones [31]. There comes the second difficulty,
0.1 D=4 related to relativistic QFT themselves. In higher dimen-
D=6 sions, normal-ordering is not sufficient, renormalization
h: φp φ−p :i/δ(0)
0.0 D=9 requires explicit counter terms, and the Hilbert space is
no longer the free Fock space [32]. Hence the Bogoliubov
−0.1 transform I proposed to adapt the states to the UV be-
havior of the free theory would no longer be sufficient.
−0.2
One would likely have to work with a more complicated
−0.3 Hilbert space.
Before these difficult questions are addressed, RCMPS
0 2 0 2 0 2 should already be applicable to a wide variety of theories
p in 1 + 1 dimension with polynomial interactions.
FIG. 2. Normal-ordered momentum space two point function
h: φp φ−p :i/δ(0) at fixed time. The results are shown for 3
values of the coupling, g = 1 deep in the symmetric phase, ∗
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